Department of Physics United States Naval Academy Lecture 32: Gravitational Potential Energy

Learning Objectives • Calculate the gravitational potential energy of a system of particles and Using the gravitational force on a particle near an astronomical body, calculate the work done by the force when the body moves. Explain the energy requirements for a particle to escape from an astronomical body.

Gravitational Potential Energy: The gravitational potential energy U(r) of a system of two particles, with masses M and m and separated by a distance r, is the negative of the work that would be done by the gravitational force of either particle acting on the other if the separation between the particles were changed from inﬁnite (very large) to r. This energy is GMm U(r) = − r

For a system of more than two particles, say three, for example with masses m1, m2, and m3, the total potential energy is Gm m Gm m Gm m U(r) = − 1 2 + 1 3 + 2 3 r12 r13 r23 Since the gravitational force ~F(r) is a variable force (its magnitude depends on r), the work done is,

Z ∞ Z ∞ W = ~F(r) · d~r = F(r)dr cosφ R R where φ is the angle between the directions of ~F(r) and d ~r. Since the gravitational force is conservative, tha work done is path independent:

∆U = Uf −Ui = −W

Force and potential energy function are related via dU(r) F(r) = − dr

Mechanical Energy: The total gravitational mechanical energy of an object of mass m GMm E = K +U = 1 mv2 − 2 r

For objects moving in circular orbits under the inﬂuence of a centripetal force (which is provided by the gravitational force), the mechanical energy becomes GMm E = − 2r This is the energy that keeps planets bound to moving in circular orbits around the sun. Note the total energy is negative, and is half the (negative) potential energy

Escape Velocity: An object will escape the gravitational pull of an astronomical body of mass M and radius R (that is, it will reach an inﬁnite distance) if the object’s speed near the body’s surface is at least equal to the escape speed. This is possible when the total mechanical energy of the object in the gravitational ﬁeld of the astronomical body vanishes - GMm E = K +U = 1 mv2 − = 0 2 r

This gives the escape speed as r 2GM v = R Notice that v is independent of the mass of the object.

(a) The ISS orbits at an altitude of 330 km. It has a mass of 419,600 kg. What is its mechanical energy?

(b) A projectile is launched from Earth and ascends to an altitude of 2REarth. What was the launch speed of the projectile? (symbolically)

(c) A projectile launches from Earth with half of its required escape speed. In units of REarth, what altitude does this projectile reach?

© 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 2.0

Two neutron stars are separated by a distance of 1.0 × 1010 m. They each have a mass of 1.0 × 1030 kg and a radius of 1.0 × 105 m. They are initially at rest with respect to each other. As measured from that rest frame, how fast are they moving when (a) their separation has decreased to one-half its initial value and (b) they are about to collide?

The three spheres, with masses mA = 80 g, mB = 10 g, and mC = 20 g, have their centers on a common line, with L = 12 cm and d = 4.0 cm. You move sphere B along the line until its center-to-center separation from C is d = 4.0 cm. How much work is done on sphere B (a) by you and (b) by the net gravitational force on B due to spheres A and C?